The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak equivalences and fibrations are detected via the forgetful functor $R-mod\to C$, right properness seems to be obvious. On the other hand, I do not understand how to check the left properness. Looking at the case of complexes of modules over a ring, I suspect that the forgetful functor respects pushouts; possibly $R-mod$ is proper if $R$ is cofibrant in $C$. Is this true? Does it make sense to pass to the category of cofibrant objects in $R-mod$?
Now a very concrete motivic question. I would like to prove that the category of modules over the Voevodsky's algebraic cobordism spectrum is proper. This assertion should probably very similar to its analogue for the case of modules over the (motivic Eilenber-MacLane spectrum) $MZ$. Yet in the paper Oliver R¨ondigs, Paul Arne Østvær, Modules over Motivic Cohomology, http://www.math.uni-bielefeld.de/~oroendig/MZfinal.pdf, I was not able to find the discussion of properness for the latter case (see Proposition 2.36). It is not stated that $MZ$ is cofibrant; one does not pass to cofibrant objects in $MZ-mod$.
Upd. It seems that Proposition 2.9 of Hovey's http://arxiv.org/abs/math/9803002 confirms my idea: everything is ok if MGl is cofibrant. Still I wonder whether the latter is known (for some model of $MGl$), and whether this result of Hovey was published somewhere.