Question: I take care of unwanted and abused pets, and occasionally I have to bathe them in a bathtub. Sometimes they go poop and it breaks up in the water so it cannot be scooped out. When this happens, I have to spray it down the tub drain with a hose.
Over time, the drain clogs, and I have to use Liquid Plumber to clear it out.

Could a shower / bathtub be connected with a direct drop like a toilet instead of a p-trap or would sewer gas come back up since there would be no water in a p-trap to stop it?

Answer: 
You need the trap, for precisely the reason you guessed. Toilets have the trap built into the toilet itself.

What you could do is get a strainer made to fit under the plug - often used in kitchen sinks. They are easy to find at most home stores or even dollar stores.

That will stop any big stuff from going down the drain, and give you a chance to pull it out.
You will need a P-Trap.

If you look at the inside of a toilet, it won't have a P-Trap inside, but it is built in such a way that gas cannot vent into the air.

[![Side of Toilet](https://i.stack.imgur.com/mV95d.jpg)](https://i.stack.imgur.com/mV95d.jpg)

It sounds to me like your issue is because of a small shower drain-line connected to your shower. To fix this, you should use a larger size pipe for your P-trap that won't get clogged.

[![Please pardon my drawing skills!](https://i.stack.imgur.com/GsWks.png)](https://i.stack.imgur.com/GsWks.png)
Please pardon my drawing skills. Good luck with your pet home!