You have to place the check marks next to them one by one. It's extremely tedious. None of this is necessary if you get a computer app like Diskaid. You just connect the phone and save the file to your computer or print it if you want.
@Europa, it was just an experiment with me. I don't save texts. But I used to could send myself a text to my email account. I was just playing around trying to figure it out. I've had very few texts I wanted to save. I just screen shot it.
Screenshot the conversation to print text messages from iPhone. Open the Messages app and select a conversation. Scroll until the text messages you wish to print are visible on the screen. Press the Sleep/Wake button and Home button simultaneously and let go immediately. Your iPhone screen flashes and takes a screenshot of the displayed messages. Repeat this process until you have screenshots of all the messages you wish to print. But it strips out useful meta information. Maybe you can try FonePaw iOS Transfer, recommended by other post.
Many ways can solve your problem.
1. You can screenshot your text messages and email to photos to your email. Then login your email on your computer and you can easily print the text messages on your iPhone.
2. Screenshot your text messages and then use air-drop to transfer the photos to your Mac. Then you can print your text messages.
3. Find a iphone transfer software that can transfer your text messages to your computer as html or other readable file formats. Use the printer connect to your computer to print text messages on iPhone.
It is such a commmon issue that most of the iPhone users face a hurdle to print text messages on iPhone. It may not able to recognize the Printer it is belonged to. They can get an actual solution from hp printer assistant to resolve their issues easily.