On contrary, the person making that claim needs to prove it's true or else we reject it as FUD against proprietary software. For a long time, the only software to resist prolonged pentesting by NSA were proprietary products certified to B3/A1 and Type 1 respectively. If NSA could hack them, they couldn't pass. Likewise, in safety, we've seen a number of high assurance systems fielded where every state (including failure) it could be in was known ahead of time. Some setups, like mainframes, hit 17-30 year uptime. All proprietary.

Meanwhile, in OSS land, we have a steady stream of easily-prevented or detected defects that compromise security. Just like in most proprietary shops. Like proprietary, those OSS projects producing highly-secure stuff are done by great designers/coders with careful review and testing processes. The community-developed OSS hasn't reached the assurance of aforementioned proprietary systems or some in academia. Yet, highly correct or secure stuff seems equally rare in both types of development with proprietary having a bit more just because there were people paying professionals to build those. In theory, with free/cheap labor, OSS could eventually produce more but there's not enough interest.

So, the endless stream of bugs in both closed and OSS software that are often really old disproves many eyes argument. It didn't work for even shallowest bugs consistently, much less deep ones. Software quality comes from people taking responsibility and putting time into QA, esp design/code review. OSS, shared-source proprietary, closed-source... always same requirement that gives quality.

GnuTLS and Heartbleed are pretty good examples for starters.