It sounds as if your puppies may have been confined too much at the breeder's and not given sufficient outside time. One seemed to make the transition to a more complex environment, the other did not. He may also not like eliminating with a person close to him (at the end of the leash). Many short coated dogs don't like to go out on wet grass - I had a miniature dachshund who did the same thing.
If you have a fenced yard let your dog out and go out with him. If not, then take him out on a leash but don't start on a walk, just keep him in a small area close by your house. Use a phrase such as "go potty" or whatever you like. Stay there until he relieves himself, and then take him on a longer walk. The reward for peeing quickly is the walk.
If he absolutely won't "go" take him back inside, and keep him on leash. Do NOT put him in his crate, as you will only be allowing the behavior to become more of a habit. Wait 3 -5 minutes, take him out again on leash. Repeat until he relieves himself outside then make a BIG DEAL out of it and do whatever he likes best - the walk, play fetch, treats, pets, all of the above.
Do NOT under any circumstances follow any advice you receive about getting a smaller crate. This will NOT solve the underlying problem and will only make things worse for your dog.
If you need more help contact Dr. Amy Marder at the Animal Rescue League in Boston.
Thank you very much. I will try this advice. I neglected to say that my dogs are 6 years old now not puppies. The breeder had them in her large mud room with a litter box in the bornor. She did not let them outside at all because they were born in November and sold in January. She felt it was too cold here in Massachusettes hence the litter box. She felt it helped the new owners and was a selling point as most were working and some were in apartments. I continued to use the litter box in my bathroom but also took them outside. I got rid of the litter box because they got bigger and would think they were all the way in the litter box when in reality only their front feet were in.
Because my experience I would strongly suggest not litter box training puppies.
I have heard of nothing but problems in both litter box and "paper" training puppies. It literally TEACHES them it is okay to eliminate indoors - and that is counter to a main basic rule of house training. So I think you are exactly right and that is why you have the problem today. The breed did not do you any favors.
I love Dr. Hetts recommendations. I think it will work if you are very consistent about it. I would also make a big deal out of it when they do eliminate on the leash. Voice and perhaps a treat praise will go along way to reinforcing the new learned behavior.