Depends on how your colleagues are, how well they mean to you and your relations with them.
Sometimes people take this opportunity to gossip or create bad image about you in the minds of seniors/bossess. Others ask because they mean well for you. Having said so, you cant really generalise the situtaion, it diifers in every situation.
If your new company is into competitve business with your old one. Its better not to mention the name. You can dodge the question, saying its part of my contract not to reveal until I join or so.
Else you could just say its a well known pharma co or mnc located in so and so place.