There is truth to that 'story' about that "Rainbow Bridge"
Will our pets be in the eternity with us? Short answer, YES..

So how do we know for sure?

Isaiah ch 11:
5 And righteousness shall be the girdle of his loins, and faithfulness the girdle of his reins.
6 The wolf also shall dwell with the lamb, and the leopard shall lie down with the kid; and the calf and the young lion and the fatling together; and a little child shall lead them.
7 And the cow and the bear shall feed; their young ones shall lie down together: and the lion shall eat straw like the ox.
8 And the sucking child shall play on the hole of the asp, and the weaned child shall put his hand on the cockatrice' den.
9 They shall not hurt nor destroy in all my holy mountain: for the earth shall be full of the knowledge of the LORD, as the waters cover the sea. (KJV)

Plus, check the last 7 chapters of Ezekiel, which describes the millennium.

And yes, of course they have a soul.

Look at Genesis:

Gen 1:20
20 And God said, Let the waters bring forth abundantly the moving CREATURE that hath life, and fowl that may fly above the earth in the open firmament of heaven. (KJV)

Gen 1:21
21 And God created great whales, and every living CREATURE that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that it was good. (KJV)

Gen 1:24
24 And God said, Let the earth bring forth the living CREATURE after his kind, cattle, and creeping thing, and beast of the earth after his kind: and it was so. (KJV)

Gen 1:30
30 And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is LIFE, I have given every green herb for meat: and it was so. (KJV)

Gen 2:7
7 And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living SOUL. (KJV)

The word in CAPS in the above verses is:

5315 nephesh (neh'-fesh);
from 5314; properly, a breathing creature, i.e. animal of (abstractly) vitality; used very widely in a literal, accommodated or figurative sense (bodily or mental):

The KJV translates 5315 into the english as: -- any, appetite, beast, body, breath, creature, X dead (-ly), desire, X [dis-] contented, X fish, ghost, + greedy, he, heart (-y), (hath, X jeopardy of) life (X in jeopardy), lust, man, me, mind, mortally, one, own, person, pleasure, (her-, him-, my-, thyself-), them (your)- selves, + slay, soul, + tablet, they, thing, (X she) will, X would have it.

So, both animals and man have a "soul".

I don't know why man teaches that animals do not have souls, and they won't be in the eternity. The Word says differently. This means that God is proud of all His creation, and man is given dominion over it. Dominion means to keep and use.

Gen 1:31
31 And God saw every thing that he had made, and, behold, it was very good. And the evening and the morning were the sixth day. (KJV)

Certainly animals are given by God for us to use/enjoy as companions and their nature is constant. All creation, including the trees and water reminds us of God in the sense that His nature is also constant. He also expects His people to be constant by the freewill He gave us.

And should you think there is no freewill I have a cat that will show you otherwise. Anyone who has ever been owned by a dog will confirm the love for their owner is unshakable.

Finally, as we know God is the same yesterday, today and forever,

Heb 13:8
8 Jesus Christ the same yesterday, and to day, and for ever. (KJV)

I can't imagine Him changing sentiment anytime.