Ziploc bags are safe for freezing food in.
I've used Ziploc bags for freezing food for years and never had any issues with the bags or the food when frozen in them.
Freezer bags are just tougher bags that are designed to hold up to freezing better than the standard Ziploc bags do.
Using freezer bags is recommended for long term storage of frozen foods in the bags but for short term storage of the foods just using regular Ziploc bags is good enough to keep the food frozen in.
I use Ziploc bags to freeze scrambled eggs and other foods in and it's never been any problem doing so.
You can also use plastic containers such as butter containers to freeze food in as well.