What kind of anti-competitive behavior might we see during this acute crisis and during the recovery?

I am a little concerned that we will see conduct and mergers that companies would not otherwise engage in, but that will try in the hopes that no one in government is paying attention, or has the interest/resources to engage in enforcement. I am hopeful that enforcers are sufficiently on the job that this won’t happen, but I am not sure.
 
Should government step in to help preserve existing companies?

There is a difference between a bankruptcy that occurs because a company no longer has a business model or product that makes sense, and one where there is a temporary problem (like no demand because of a pandemic) but sound fundamentals. In the former case, the assets should be redeployed to their best use, and there is no reason for the government to stop that from happening. 

In the second, the best use of the assets is what they are doing—and while the economy is frozen, there isn’t anything else for those assets to do, either. In that case, everyone is best off waiting for economic activity to restart. The way to do that is to restructure the finances of the company but not change any real resource allocation. Lenders and owners and managers understand how to do this, but the government can still help by creating laws and resources that help reduce transaction costs and give bargaining power to the right parties.

Are the incentives of the healthcare industry well suited to a crisis like this?  

No. Incentives for innovation in U.S. healthcare are set up to be slow and to generate slightly better—and much more expensive—innovation. What we need now is far faster and cheaper innovation, even if it is somewhat lower quality, e.g., a ventilator with fewer settings or a less convenient user interface. The crisis really brings out this mismatch. But when people’s lives are at stake, a different set of innovators arrives (e.g., MIT’s respirator project), and the FDA becomes more flexible because they have no choice. So we can hope!