Stopovers on Shinkansen services?

As macraf notes in his answer, all Shinkansen services north of Morioka (Hayabusa and Hayate) have only reserved seats. Between Morioka and Tokyo, you can take a slower Yamabiko train, which will have unreserved seating.

If you want to exit the ticket gates at the intermediate station

This is the JR definition of a "stopover" (????). Basically, base fare tickets allow stopovers (unless it's for a very short trip inside an urban area) but limited express tickets (either for reserved or unreserved seats) do not.

So in this case the best strategy is to buy a single base fare ticket from Hakodate to Tokyo, and two separate limited express tickets. When you exit at your intermediate station, your base fare ticket will be returned, and you can use it again to re-enter. Also, since this is a very long trip, your base fare ticket will actually be valid for several days (the validity period will be printed on the ticket), so you could even make several overnight stopovers.

If you do not exit the ticket gates at the intermediate station

If you are using reserved seats, the above also applies because reserved seat tickets are only issued for a particular train on a particular day, and are not valid on any other train.

If you are using unreserved seats, you can get off a train and on another one with a single limited express ticket because unreserved seat tickets are valid on all trains on a particular day. (For example if I buy an unreserved seat ticket from Morioka to Tokyo valid today, I can stop in Sendai for a bowl of udon, and again in Koriyama to buy the excelent senbei they sell there.)

So in your case, for maximum stopover flexibility you can buy a reserved seat ticket from Hakodate to Morioka (since trains on this section have no unreserved seats), and then an unreserved seat ticket from Morioka to Tokyo (and you can make as many stopovers as you want on this section as long as you don't exit the ticket gates).