How does pre-training help with semantic segmentation with U-net
The procedure you described is how to do fine-tuning of a network for object classification.  There are two standard techniques: (1) freeze all but the last few layers, replace the last layers with new layers of the appropriate size and random weights, and train them; or (2) train all of the layers, starting from the weights of the pre-trained model as your initial weights, but use a very low learning rate.  You can also do (1) then (2).  See https://datascience.stackexchange.com/q/28383/8560 for a nice overview.

I'm not an expert on u-nets, but it appears they are used for image segmentation rather than object classification.  Therefore, the same approach of freezing the last few layers probably isn't suitable here.  You can still use method (2), i.e., fine-tuning using stochastic gradient descent with a very low learning rate.  If your task is similar enough to the task the pre-trained network was trained for, that might be effective.

I can't see any way to use a pre-trained model for object classification (like VGG16), as they use a different architecture.  Rather, you'd probably have to start with a pre-trained model for segmentation (i.e., a pre-trained U-net model).  The U-net paper says that they make pre-trained models available on their website, which perhaps you could use as a starting point.