Yes they do, when you are in a high demand industry with low supply of skilled workers, you are in the driver's seat of a negotiation. You have options. You're the one being bid on by multiple employers. You absolutely have the leverage there. This is similar to the real estate market, by the way. There are so-called "buyers markets" and "seller markets" where one trend or another is dominant at any given time depending on if there are more buyers than sellers or more sellers than buyers.
It's not easy to replace tech employees at all. Even if there was an abundant supply of expert software engineers (there's not), it takes many months to ramp up on a new code base to optimal productivity and to learn the organization and domain requirements of a software product. That will never change.
An average tech company will spend tens of thousands of dollars to fill a position, not even counting initial salary and bonuses.
Leaving for somewhere better often does solve quite a bit. On my last job move my pay increased by 50%. If I had stayed at my previous company I'd be skating by with 4% raises.
On the whole the aggregate effect of employees moving around in a dynamic market creates the cush benefits and pay we see in this industry.
The reason they wouldn't give you specifically better insurance is because they have to negotiate a plan for everyone in the company. What is that costing them on paper? I don't know how many employees your company has or what their margins are or any details really, but it can be as much as $700-$1000 a month per employee for health insurance. If you've got 100 employees that's $100K a year.
No boss should "laugh you out of the office" for requesting that. But really, its an HR and Corporate leadership issue so you should take your grievances to those with the power to make the change and not your boss (unless you work directly for the CEO).