Not much in the way of dining on Singer Island. Best bet would be to drive north on the island to US 1 and head north--There are a number of seafood places heading up toward Juno Beach and Jupiter (all on the east side of the road). Tops would be Little Moir's Food Shack (SE corner US 1 and Indiantown) in a strip mall in Jupiter and Captain Charley's Reef Grill in Juno Beach, followed by Juno Beach Fish House and the Hurricane Cafe, both near Donald Ross Road.
I'm not sure about vegetarian places in the area--one possiblity might be Off the Vine, on PGA Blvd. in Palm Beach Gardens-when you leave Singer Island, at US 1, continue staight onto PGA Blvd for a few miles, past Military Trail. Another spot might be the Cheesecake Factory in the new Downtown at the Gardens, off PGA--great salads, so large you can share them. Hope this helps.