Hi Jayme -- Yes, junipers can bounce back from browning caused by dog urine, especially if the browning is not severe. Dog urine can cause foliage to appear scorched and may lead to the needles turning yellow and then brown before they drop from the plant. Evergreens also typically shed their oldest, innermost needles every year. Dy conditions could have some effect, too.   

If the browning is due to dog urine, rinsing the foliage with water to wash away the salts can help. Water deeply through spring (making sure it drains well). In June check to see where the shrub has greened up. If the tips are green, the branches are fine. Don't prune back from a green tip into the brown area on junipers if you want that branch to continue to grow. Those internal brown needles will fall off on their own. If the tips are not green, then prune back to the center stem just to the branch side of its attachment to the stem. It can take some time for a juniper to recover, depending on the extent of the damage.

Do keep the dogs from urinating on this if you can until it's more mature, just to be on the safe side. Good Luck!