MSE to predict Values 0 and 1
Predicting $0.5$ for all items in your case would also give you MSE of $0.25$. That is because independently of whether the true label is $0$ or $1$, the squared error for each example will be $0.5^2 = 0.25$

Your model is performing badly under an MSE measure, when such a simple model that does not take input data into account can get the same score.

However, it is debatable whether MSE gives you a useful metric here.