If I understand your situation correctly, the answer is yes, the cube lemma holds, but you cannot use that to get a full model structure. Your situation often arises when trying to transfer a model structure from a cofibrantly generated model category $M$, along an adjunction $F:M\leftrightarrows N: U$, to a bicomplete category $N$, e.g. to a category $N$ of algebras over a monad. Check out Lemma 2.3 in Schwede-Shipley *Algebras and Modules in monoidal model categories*. For them also, the only difficulty is proving that the pushout is a trivial cofibration in $N$ (defined via $J$-cell) is a weak equivalence. The good news is that, even without this condition, you can probably prove whatever you want. Specifically, the structure on $N$ may be that of a *semi-model structure*. Have a look at Definition 2.3 in Spitzweck's thesis. Note that on page 14 he remarks that the cube lemma holds in this setting. Indeed, Karol's answer is spot on: in a semi-model category, pushouts of spans $A\gets B \to C$, where $B \to C$ is a trivial cofibration and $A$ is cofibrant, yield trivial cofibrations.

Even if your $N$ is not arising via a transfer along an adjunction, it may still be a semi-model category as defined by Benoit Fresse in 12.2.1 of this book. Again, the cube lemma holds. In general, any statement about model categories holds for semi-model categories if you restrict to the subcategory of cofibrant objects, or cofibrantly replace everything in sight. Since the cube lemma is already about cofibrant objects, there is no problem.

Unfortunately, it does NOT follow that every semi-model category is a model category. An example is the category of non-reduced symmetric operads in $M = Ch(\mathbb{F}_2)$. As a category of algebras over a $\Sigma$-cofibrant colored operad $P$, it is a semi-model category, by Spitzweck's Theorem or by Fresse 12.2.A applied to the transfer from the category of collections $\prod_{n \in \mathbb{N}} M^{\Sigma_n}$. However, it is not a full model structure: the pushout of $P(0)\to P(K)$, where $K$ is an acyclic chain complex $C$ in level zero and 0 otherwise, along the map $P(0)\to Com$ to the terminal non-reduced symmetric operad, is not a weak equivalence, because it introduces a summand of $(C\otimes C) / \Sigma_2$.

My advice: stick with the semi-model structure and use it to prove whatever you need. This was the approach taken by Spitzweck, by Goerss-Hopkins in their obstruction theory paper, by Fresse, by Hovey in his paper *Monoidal Model Categories*, and in many of my papers. A semi-model structure is good enough.