Cyclocross vs road bike I am thinking about getting a new bike. 

I have a MTB but I am sort of petrified of road riding (cars pass way too close). I have found a nice Specialized Cyclocross bike. 

I wanted to have a bit more practice and I do own a Trainer but can I fit a cyclocross on a trainer?
Most trainers on the market today can do 26", 29" and 700c. Specialized's cyclocross bikes are 700c, your trainer will likely be able to handle it fine, but check the manual (*).

A trainer won't help you gain confidence riding in traffic though -- you'll just have to practice riding in traffic, which is the bigger problem here. 

(*) Your trainer may have specific recommendations, e.g. no carbon fiber bikes in the trainer or something. Read the documentation.