When a couple eats at a restaurant, and a man lays his napkin on his plate, does that prompt the waitress to ask to take away both plates?

Putting the napkin on the plate when finished means he’s insulting the chef.  

Wait staff look for the position of cutlery, so if you’re not finished don’t lay your cutlery neatly on the plate cross them or keep the fork in your hand.  When attending the table to remove the dish you were eating from they will not ask if your finished, they will ask if they can remove the plate.  If there is an empty glass or plate without cutlery on the table, they may ask if they can be removed while you are still eating 

Educate your husband. 

In scouting, we learn the rule "The slowest hiker sets the pace for the hike." Along with that we learn "Someday when the slowest hiker is YOU, you'll be glad for this rule." This rule applies to restaurant dining as well. 

Playing at "I'm all done and waiting on you, slowpoke" is thoughtless and rude. Your husband should not be finishing well ahead of you any more than a faster hiker should leave behind a slower hiker. Your husband can eat quickly if he wishes. But when he's finished eating he should continue to pick up his fork and push food around on his plate to give an appearance that he isn't yet finished. 

Servers respond to his "I'm finished" signals because they don't want to risk ignoring such signals from a person who may be paying the bill, leaving the tip. Some inconsiderate diners wouldn't hesitate to tell the server "Yes, we're finished" without regard for YOU not being finished. And the server's job is to satisfy the paying customer, to NOT notice that the paying customer is a being a boor. 

Miss Manners tells us that one signals "finished" by laying ones fork and knife on the plate parallel, usually in a 9 to 3 o'clock or 11 to 5 o'clock position. Laying the utensils on the plate in a crossed position signals "not finished, just pausing." 

I've never heard of laying the napkin on the dirty plate. Sounds kind of yucky. 

Effective wait staff can read their diners manner/body language to ascertain what next step to take in terms of taking an order, clearing a table, refilling the water and the like.

just tell them youre not done if they ask you

When he's finished eating, he should lay his napkin on the table beside his plate, NOT on the plate. It doesn't matter whether you've finished eating or not, but obviously he shouldn't look up and make eye contact with the waiter, or they'll naturally come over and ask if they can clear the table.

The waiter should NOT approach the table if one of you is still eating. But you can reduce the chances of this happening by making it  clear that you ARE still eating. If you lay your knife and fork down and look around or start chatting, they might not notice you're still chewing, and it would then be natural to come and ask whether you've finished.

Your husband should keep his napkin on his lap until you're both finished.  Then, put the napkin next to the plate, not on the plate.

No it doesn't. She waits for both napkins.

Your husband should stop putting his napkin ON the plate.

We servers HATE that.

If he's finished, he can put his napkin on the table to the side of the plate.

If the server asks if you're finished, just say, "No, as you can see, I'm still eating".