There is no indication that animals have eternal souls. However, God is perfectly capable of recreating (not the same as resurrecting) an animal who has died. There's no Scriptural proof, but I wouldn't be surprised if He does that.
For more information on the subject of Heaven, see Randy Alcorn’s book Heaven.
Randy Alcorn (@randyalcorn) is the author of over sixty books and the founder and director of Eternal Perspective Ministries.