Is it necessary to take log transformation on the data values to get the minimum mean squared error?
If you applied the log transformation to all your labels as well as the inputs, then it is normal that the MSE decreases. It is the case because when you apply log to big numbers they all become smaller and less spread. The standard deviation of the log price is a lot smaller than the one of the price. The MSE of any model trying to predict the log price (for instance the model that predicts the mean log price every time) will be smaller than the MSE of a model trying to predict the price.
Using the log square footage to predict the price could be useful if you are using a linear regression for instance and if the relation between square footage and price were logarithmic. It is probably not the case but you can try it.
However changing the target (applying log to it) changes the loss (you are computing the MSE between log price and prediction) and thus changes the problem you are trying to solve. You cannot compare the performances of the two models directly using the outputed MSE.