Yes but it is best to make it a head shot, or the squirrel will suffer and die.
The Crosman 1088 Pellet/BB is rated at 400-430 FPS
See the related link at the bottom of this page to see the Crosman 1088 parts manual.
see the link below
see the link below for the owners manual
It would really help if you stated the manufacture of the 1088. So I'm going to guess it's a Crosman. See the link below
Crosman has a web page for their owners manuals and part list. See the attached web page. you will find your 1088 listed about half way down on the page. The parts last will give you an exploded view of the gun. See the related link
see the link below for the Owners manual.
yes a squirrel can kill a baby chicken
It's hard to help you, considering you did not say what is wrong with your airgun. So here is what I suggest. Crosman offers a service to locate shops that repair older Cromann products. See the link below.
If it's not on the outside then very possibly it's under the grip cover
Possibly if it stood on the squirrel or headbutted it into a tree.
It is only legal to kill a squirrel for hunt in the spring season. This would be known as squirrel season and small game is fair game. If you accidentally kill a squirrel in a non spring season it will be ok because you are not intentionally hunting them.