Well, it is possible to put one on your rifle. A 30-06 is a bit of an odd caliber to want a muzzle brake on but that's the kind of job a good gunsmith can do fairly easily, be sure to call one to ask for a general estimate on that job before deciding.
Since the .308 and 30-06 bullets are the same .30 caliber diameter you could merely have a .308 caliber muzzle brake fitted to the rifle. Usually this is either done by pinning the muzzle brake on or threading the actual barrel itself to accept the threads of the muzzle brake.
It costs extra to have the muzzle brake blended with the barrel to not have any rough edges and look more natural between the brake and the barrel and usually there is a bit of refinishing involved with either of the processes.
The reason you dont see many muzzle brakes on hunting rifles is due to the restrictions of the ammo allowed in the magazine of a hunting rifle, thus reducing the need for quicker follow-up shots. On the other hand, a tactical rifle like the DPMS LR-308 or M1A Springfield (which both hold more rounds than a hunting rifle)is often used in a life threatening situation by LE or military persons with the possibility of multiple threats so a muzzle brake is practical for quicker follow up shots and less fatigue in longer shooting sessions at the range at the expense of a louder muzzle report.
Of course there is always the exception to the norms, I saw a fella with a muzzle brake on his .17 HMR bolt action rifle. Since most can handle the .17 HMR recoil, you might ask why he would want one. He wanted a rifle that would look cool and drive bullets through the same hole as quickly as possible as he could with a bolt action. His gun was also in the shop for precision bedding after the muzzle brake was fixed.