The answer is yes - at least if you take for the definition of $HH^*$ the self-ext of the identity functor.
For any quasicompact quasiseparated scheme we know (thanks to Thomason-Trobaugh) that $D_{qc}(X)$ is compactly generated by the perfect complexes. This means that $D_{qc}(X)=Ind(D_{perf}(X))$ -- the quasicompact [dg enhanced throughout!!] derived category is the ind category (in the $\infty$-categorical sense) of the perfect one.
Now it's an easy consequence of various results in Lurie's Higher Topos Theory [and maybe by now appears in there explicitly?] that
$\bf Proposition$: Passing to ind-categories induces an equivalence of [symmetric monoidal] $\infty$-categories between small, idempotent-complete stable $\infty$-categories (with exact functors) and compactly generated presentable stable $\infty$-categories (with proper functors -- continuous functors that preserve compact objects).
(This is spelled out in my paper with Francis and Nadler.) This is very useful -- it means you can go back and forth freely between small concrete categories that you like and big flexible objects which have all (small) limits and colimits, where the adjoint functor theorem and many other things apply.
So the passage from perfect to quasicoherent is part of an equivalence of categories.
It also means in particular that all exact functors on perfect complexes are representable by quasicoherent sheaves on the square --- though unless you're smooth and proper these will not be precisely the perfect complexes on the square..
Since Hochschild cohomology is given by self-Ext of the identity functor (which is certainly a proper functor), we can choose to calculate it either with the small categories or with the large categories. (For a reference for the equivalence between dg and stable $\infty$-categories I suggest this paper by Blumberg, Gepner and Tabuada.)
So now you might want to check that the cyclic bar construction for a small dg category does indeed calculate self-Ext of the identity functor. (Maybe you want to take the dg category to be pretriangulated, or first prove that both sides are invariant under Morita equivalence.) This is pretty clear I think - at least notationally it's easier if we assume our category has one generator (equivalently finitely many -- and this is always the case for [q-c,q-s] schemes), hence is just modules over a dg algebra. Then we observe the standard bar construction is a free resolution of the algebra $A$ considered as a bimodule (i.e. of the identity functor), and the cyclic bar construction computes the Ext. For the multiobject version I think there's an MO discussion already somewhere..
There are conceptually simple definitions, but they require a more symmetric definition of Hochschild homology. The Hochschild homology of $X/k$ (with coefficients in $\mathcal O_X$) is the homology of the complex
$$ \mathbf R\Gamma(X\times X, \mathbf R\Delta_*\mathcal O_X\otimes^{\mathbf L}\mathbf R\Delta_*\mathcal O_X)). $$
Equivalently, in the language of derived algebraic geometry, this is the complex $\mathcal O(LX)$ where $LX=X\times^{h}_{X\times X}X$. The circle comes in because $S^1$ is the homotopy pushout $*\coprod^h_{*\amalg *}*$, so, formally, $LX=X^{S^1}$ is the free loop space of $X$ (it is a derived scheme whose underlying scheme is $X$). Now $S^1$ is also a group and it acts on itself, hence it acts on $LX$ and on the complex $\mathcal O(LX)$ computing $HH_*(X)$.
The complexes computing the cyclic, negative cyclic, and periodic cyclic homology of $X$ are respectively the homotopy orbits $\mathcal O(LX)_{hS^1}$, homotopy fixed points $\mathcal O(LX)^{hS^1}$, and Tate fixed points $\mathcal O(LX)^{tS^1}$ of this circle action. (For $G$ a compact Lie group acting on a spectrum or chain complex $E$, there is a norm map $(\Sigma^{\mathfrak g}E)_{hG}\to E^{hG}$ whose homotopy cofiber is by definition $E^{tG}$. In our case this cofiber sequence induces the usual long exact sequence relating these three homology theories.)
I'm not sure what the cyclic cohomology of a $k$-scheme is. At least if $X$ is affine, it is computed by the dual of the complex $\mathcal O(LX)_{hS^1}$, i.e., by the complex of $S^1$-invariant maps $\mathcal O(LX) \to k$.
ETA: Hochschild cohomology of $X$ with coefficients in $\omega_X$ is computed by the complex $\omega(LX)$, which has an $S^1$-action. So perhaps one gets reasonable "cohomology" versions of the above theories by replacing $\mathcal O(LX)$ by $\omega(LX)$.
To relate this to the traditional definitions, one shows that there is an equivalence between $S^1$-equivariant chain complexes and mixed complexes, and that the $S^1$-action on $\mathcal O(LX)$ corresponds to Connes' operator $B$.
Here's yet another way to understand the $S^1$-action on Hochschild chains which applies to the noncommutative setting as well. The complex $\mathcal O(LX)$ can be identified with the Euler characteristic (=trace of the identity) of $D_{qcoh}(X)$ in the symmetric monoidal $\infty$-category of presentable dg-categories. It is a general fact that the Euler characteristic of any object comes with an $S^1$-action. From the point of view of the cobordism hypothesis, this $S^1$ is now the framed diffeomorphism group of the circle, which is the Euler characteristic of the universal dualizable object in $\operatorname{Bord}_1^{fr}$.
If $\mathbb Q\subset k$ and $X$ is smooth and affine, the relations with Kähler differentials are given by:
\begin{align*}
HC_n(X) &= \Omega^n(X)/B^n(X) \oplus \bigoplus_{i\geq 1} H^{n-2i}_{dR}(X),\\
HC_n^-(X) &= Z^n(X) \times \prod_{i\geq 1} H^{n+2i}_{dR}(X),\\
HC_n^{per}(X)& = \prod_{i\in\mathbb Z} H^{n+2i}_{dR}(X).
\end{align*}
(Reference: Loday's book, 3.4.12, 5.1.12). The formula for $HC^{per}_n(X)$ remains valid if $X$ is not affine (because it satisfies Mayer-Vietoris), but the first two become more complicated...
Update. Here are the global formulas, I hope I got the indices right:
\begin{align*}
HC_n(X) &= \bigoplus_{-\dim(X)\leq i\leq n} H^{n-2i}_{Zar}(X, s_{\leq n-i}\Omega^*_X),\\
HC_n^-(X) &= \prod_{0\leq i\leq \dim(X)-n} H^{n+2i}_{Zar}(X, s_{\geq n+i}\Omega^*_X).
\end{align*}
Here $H_{Zar}$ is Zariski hypercohomology, $s_{\leq k}$ and $s_{\geq k}$ denote the stupid truncations, and $\Omega^*_X$ is the de Rham complex.
Best Answer
HKR is true in characteristic $p>0$ as soon as you assume that $p$ is greater than the dimension of $X$.
You can find a very nice proof of that in a recent preprint of Arinkin-Caldararu: http://arxiv.org/abs/1007.1671
I am sure this is not new, but the reason I am quoting this paper is that you will find in it a very nice generalization of HKR for closed embeddings $X\subset Y$ (where the result works in positive characteristic whenever it is greater than the codimension). The Hochschild case is the diagonal inclusion of $X$ into $X\times X$.
EDIT: I am not an expert in algebraic geometry, but I believe your question is related to Deligne-Illusie proof of Hodge-to-de Rham degeneration, which has the same kind of restriction (p greater than the dimension).