Not everyone–especially a Veteran–feels comfortable saying they’re struggling. No might mean no. Or it might mean, “No, but I’m getting there.”
Also be prepared that you might just get “I’m fine. I’m dealing with it”, which doesn’t necessarily mean no. Mention specific things that have you concerned in a caring, non-confrontational way.
If you feel like you might be missing part of the picture, remember that others care for the Veteran too. Think through how you can learn more from others in their life.