The answer is 'no'. The loglinear model is more general than the logistic regression model. See Fienberg, 1980, Analysis of Cross-Classified Categorical Data, section 6.2 on how to specify a loglinear model so that it corresponds to logistic regression.
Actually the reverse is true: If all variables are categorical, then every logistic regression model corresponds to some loglinear model.
As written, your question can't work, since y is a 0-1 variable and you're doing logistic regression.
If you mean that the linear predictor had a nonlinear relationship with one of the independent variables, that is, $\eta = a + bf(x)$, say, for some nonlinear $f$ (with all other variables held constant), then you can write $x^* = f(x)$ and put $x^*$ in your logistic regression as an independent variable. [In a logistic regression, $\eta = \text{logit}(P[Y=1])$]
This is quite commonly done in linear models and generalized linear models; there's a linear relationship, but it's with a transformed independent variable. Under the usual assumptions you need for a GLM, the transformed variable works perfectly well as a predictor.
Note that if $f$ is known and that coefficient, $b$ is known, you don't put $x^*$ in as a predictor, because $x^{**} = bf(x)$ is then an alternative predictor with coefficient 1; those come in as offsets (e.g. specified in R by using the offset argument). (In ordinary regression you could let $y^* = y-x^{**}$ instead, for the same effect.)
I will assume the coefficient of $f$ is unknown (though you specified it to be 1).
In your particular case $x^* = f(x) = (x-4)^2$. If you were unsure about the "4" there (e.g. if it's just a rough guess or something, rather than a value that's definitely known), then you could instead use two new variables, $x^*_l = x-4$ and $x^*_q = (x-4)^2$ both as predictors, which will capture a general quadratic relationship (with the additional benefit that if the '4' is nearly right, the estimates be nearly uncorrelated with each other and with the intercept.
Best Answer
Recall that the Logistic regression model is a non linear transformation of $\beta^Tx$
So to answer your question, Logistic regression is indeed non linear in terms of Odds and Probability, however it is linear in terms of Log Odds.
A simple example
Fitting a logistic regression model on the following toy example gives the coefficients $\alpha = -5.05$ and $\beta = 1.3$
Plotting the probability $P(Y=1)$ as a function of $X$ clearly shows the non linear relationship
The Odds of $Y$ being 1 given $X$ is also non linear
Finally the log odds of $Y$ being 1 is a linear relationship
See here for some more details: Calculating confidence intervals for a logistic regression