Can a credit card company have you arrested for not paying them? I live in Texas and we have not debtors court or garnishment, I know they can put a judgement against me but they have served me with papers for a court hearing.

As long as you made the charges I suggest you pay the bill. You can hurt your credit score greatly by not paying them. I'm not sure if they can arrest you in Texas, but try paying it off in small amounts.
Did you make the charges?

I tried to work out payment arrangements with them at 300.00 a month and they refused. I have tried over and over to work out arrangements and they will not do it.

No, they cannot have you arrested. Debtor's prison went out bu the 19th century. But they can attach bank accounts. If they are suing you, they seem to think they can do better in getting a judgement then $300/month.

They may sue you in civil court, so yes you can and may go to court on it, They do not allow garnishments in TX but they can get an attachment of your bank accounts,