i had a similar issue with my civic '92 , and after a couple of months of head ache i found that the main relay had one pin detached partially , so all i had to do was to take the relay out and re-solder that pin , you could barely see that the solder wasn't good so look very closely when you take it out . When the car was getting hot in the summer especially i was getting stuck at gas stations and anywhere on the road because i wasn't getting any fuel .After a few minutes (hours sometimes) it would start again. The main relay is located under the steering wheel at civics and i believe it's the same for integras .Try to pull it out for a visual inspection.
There many other possibilities i'm sure but try this one and let us know if you fix it.