Lately I seem to have a problem with my SPD clips squeaking. Have tried lubricating them and that seems to quiet them for the first 20 miles and then they start in again, particularly when I am bearing down hard as on a hill, etc. Anyone else have this problem? Otherwise, everything seems tight and functioning well.

I had this problem with my old bike last year. I finally determined it was my left pedal/cleat that was squeaking and if I changed the angle of my left heel slightly the squeak would go away, until I forgot about maintaining that angle and then it would come back.
Replaced the bike this spring and swapped the pedals over to the new bike. After a couple of rides my squeak was back. Like you it took a few miles of riding before it started.
I finally got tired of it and decided to figure out if it was the pedal or the cleat. I swapped the cleats from right to left and I haven't squeaked in a couple of hundred miles now.
When I had both of the cleats off I examined them closely and although they both showed some minor wear there wasn't anything obviously more worn on the left than the right.
For now I'm happy, no squeaks and if and when it does come back I'll probably just see about getting a new set of cleats.

Poor cleat alignment, either rotation or tilt, is a common cause of squeaking. Also excessive wear.

Lack of oil between the toes

Diving me nuts as well. I purchased a pair of Shimano SM087 Mt. shoes last season, hoping the stiffer sole would help eliminate hot spots.

While the hot spots somewhat got better, the shoes started squeaking after a seasons use. With any of 3 different Shimano SPD pedals. I lube the top and springs of the pedal as often as I can remember and sometimes get thru a 3 hr ride with no noise and sometimes not. The ***** is, the shoes fit really well and I have 4E feet, but I never had the problem with my older Diadora shoes nor with my Diadora winter shoes.

As a result of the hotspot not really improving as I had hoped, I may shortly be investing in Shimano 105 level road SPD's and a new pair of shoes, under the theory that a wider cleat and platform will help maybe reduce the hotspots as well as the squeaking.

And FWIW, it's not a cleat alignment issue as these cleats are setup identical to other shoes. I also just yesterday replaced the cleats. Squeaked all morning on today's ride

Sometimes it isn't the cleats, it's the pedal hitting the shoe itself.
I had some Shimano shoes that were driving me nuts on Coda (Cannondale) pedals. After forever I realized it was the shoe sole.
I hacked a bit away where it met the pedal, and all was better.

I oil the springs in the pedals with triflow and that usually fixes the problem.

I suspect that is exactly what I'm hearing - rubber sole on the pedal, nothing to do with the cleats at all.

My concern with shaving down rubber is aggravating the hot spot issue by reducing the amount of area the shoe contacts the pedal. You would think that Shimano shoes with Shimano pedals would not have this issue.

Being as it's road bike only, I may very well go the road pedal/shoe route.

Called the bike shop and was advised to use some waterproof grease on the springs, etc. Considering the fact that the squeaking stopped when I used a thinner lubricant and only started after 20 miles I think I agree with the mechanic at the shop that the lighter oil dissipated and the grease will probably solve the problem. We'll see. Of course I wiped away the excess and tried to work the grease into the spring mechanism. Was also suggested that I remove the cleats and apply lubricant between the cleat and the shoe but I would rather take the easier course first.

I have used a dry, wax based chain lubricant( White Lightning) on SPD latching points and cleats. It seems to work OK.