I would disagree with Laura's advice about allowing your communication system to operate in-tandem to what you're trying to displace. Hospital staff have very little appetite to learn or adopt new systems, especially for mission-critical work. I also disagree with the idea that your competitors are an ideal partner. They will force you to adopt their processes and live by their rules, severely limiting your upside and will often waste your time with misleading signals of interest.

Has this objection cost you any sales yet? If not, don't over-think this. Is it an objection that you've just been given and you're about to close or lose a sale over? If that's the case, call me or someone and talk through this right now.

Otherwise, focus on ensuring that the client is totally bought-in to using your software (trial or otherwise). If this issue comes up, you have a number of ways to address this:
Generally speaking, even with mission-critical software, the due diligence doesn't include financial analysis of the vendor. So just project the confidence that you *will* be around and you should be fine. If not, there are actual contract-specific language that you could use to address this but you really should avoid that at this stage.