You can ask, but be aware that in this particular situation they would likely say no. You gave them a start date, now you want to change it for what business is going to view as a frivolous reason.
The business has needs too and they likely want you to start on this date. Since this is entry level, they may be starting a group together and providing some training or orientation that they want to do as a group. Or they may have contractual reasons why they have to have you on board by a certain date.
Just asking for a later date after you have accepted an earlier one can get your offer removed in many places because you are now viewed as unreliable. This is a risky move. It can work out, but you are taking a chance that they will decide against you. After all there are lots of entry level people looking for jobs and they may have a second choice who would be willing to come when they want him. At entry level, you have very little leverage in a negotiation. So at least if you do ask, make sure they know you are still willing to come in on the originally agreed date to reduce the risk of them saying, "no thanks."
For the future, If you want a break between jobs, then state this at the time they ask you about a starting date. Never tell them one date and then try to change it later unless a personal emergency has come up.