Edit: Yes, the number of positive or negative eigenvalues in $AB$ and $B$ are the same. The spectrum of $AB$ is identical to the spectrum of $\sqrt{A}B\sqrt{A}$ and by Sylvester's law of inertia, $\sqrt{A}B\sqrt{A}$ and $B$ have the same number of positive/negative/zero eigenvalues.
To address the problem that the OP actually wanted solved:
Given two symmetric positive definite matrices $C$ and $D$, show that $CD+DC$ is also positive definite.
This is not true, as can be seen from the following example:
$$C = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}, \quad D = \begin{bmatrix} 100 \\ & 1 \end{bmatrix}.$$
Then their respective spectra are:
$$\sigma(C) = \{1,3\}, \quad \sigma(D) = \{1, 100\}, \quad \sigma(CD+DC) \approx \{-20.2724, 424.272 \}.$$
The reason is simple: when computing $\det(CD+DC)$, the big factor $100$ participates in the positive part only $2$ times, but squared in the negative (counter-diagonal) part.
This might give you some insight how things behave in a more general setting. For example, I'm fairly certain that it can be shown that the following is true: if $C$ and $D$ do not commute, there exists $n \in \mathbb{N}$ such that $CD^n+D^nC$ is not positive definite. All you need to do is observe a $2 \times 2$ submatrix for which $D$ is not a multiply of $I_2$ (WLOG, you can assume that $D$ is diagonal). In other words, any such $D$ can be blown enough to make the above invalid.
This would give you that for every symmetric positive definite $C$ (such that it is not a multiple of identity), you can find a symmetric positive definite $D$ (diagonal, if you want) such that the above does not hold.
Best Answer
If $x^TAx>0$ for all $x\neq 0$, then $x^T(-A)x=-x^TAx<0$ for all $x\neq 0$. So yes, if $A$ is symmetric positive definite, then $-A$ is (symmetric) negative definite.
The cheapest way how to check whether a symmetric matrix is positive definite is to compute the $LDL^T$ factorization of $A$. If all the pivots on the diagonal of $D$ are positive, then $A$ is SPD. This test is much cheaper than to compute all the eigenvalues of $A$.