It is nearly impossible to replicate restaurant style hash browns. They are steamed, peeled, steamed again or boiled, shredded and cooked partially in oil, oven dried and IQF before going into a bag to be cooked at home once again. IQF equals individually quick frozen strands. Cooking the potato twice is the key and is something that is hard to replicate at home.

Some restaurants that do fresh hash browns bake potatoes, then cool, peel and shred. The problem you are running into is that the potato is 80% water. So, squeezing gets rid of some of the moisture, but not all.  Look at recipes for latkes and you’ll see they sometimes add flour to dry out the potato mixture a little bit before placing in the fry pan.

 Here is another possibility: How Do I Make My Own Freezer Hash-Browns?

By the way, use very little oil or make sure the oil is hot and spread out the shreds of potatoes so they are not all grouped together or they will absorb the oil and will just steam in the pan.

Hope these suggestions help.