Let me approach that question this way: the answer is partially yes. The committees of the House, such as this committee, are fully covered by privilege, regardless of what they're talking about.
With the board, it would enjoy the protections and privilege according to what it's talking about. The privilege it would have the protection of is the privilege the House enjoys to administer its own internal affairs. It can administer its internal affairs publicly or privately, but if it's a matter within the ambit of that privilege of managing its internal affairs, then it would have privilege.
I can only remind you of the decision in 2005 of the Supreme Court of Canada, where the House took the position that administering all its employees was within its internal affairs. The court said no, it didn't go that far. In other words, how the House might deal with the clerk as an employee is one thing, but how it might deal with somebody working outside on the grounds might be another. The court didn't really draw the line as to where the line is, but somewhere in there is a line.
In a similar fashion, the board would have to address its mind to what is the ambit of internal affairs. My own hunch would be that, by and large, most of the business discussed by the board would be within that privilege.
I would just add, if I may, that the absence of privilege, in my view, should not discourage the board from meeting publicly, insofar as the board is a body created by statute. It's not a parliamentary committee. Its responsibilities relate to the financial and administrative affairs of the House, which is a public institution. So, arguably, the matters it's discussing are not ones that need to have the protection of privilege, such as free speech, which of course is designed to enable members in debates to be able to speak freely and not to be impeded in doing so by legal considerations about defamation. The same, arguably, wouldn't apply to the board, so there may be less need for concern about privilege in a public meeting of the Board of Internal Economy.