can I run 6/3 from a 60amp breaker on my meter loop to my shop 50ft to supply 120v/220v
Yes #6 wire would be fine on a 50 amp breaker. Today’s code requires 4 wire and you will need a grounding electrode at the the shop.   I would verify that the original #4 is not serviceable prior to spending the $ on a new UF cable or putting in conduit and wires, over the years I have found several feeders like yours that were cut because the owner was afraid of 220/240v. Most don’t realize the voltage to ground is only 120v there is only 220/240 between the 2 hot’s. I had a friend do the same because he did not understand and realized his welder would not work and asked me to fix it, prior to that I wondered why someone would do this (I have seen it about 5 times) where there was no damage to the feeder.