For the content that your raiding guild is working on, raid composition isn't that big of a priority. This isn't the age of The Burning Crusade or Wrath where healing classes had specific roles. Cataclysm has helped equalize the playing field somewhat. It's the whole "bring the player not the class" mentality. Every class has the tools and capability to cover the raid or cover the tanks effectively. So, it's up to the players to see what they're good at and what they prefer to do. Some healers prefer staying on one tank only. Others enjoy the challenge of looking after everybody. Granted, it feels to me that some classes have a slight edge over others depending on what is being healed. But that's a topic for another time. I would say that raid composition matters when you get to increasingly challenging content. Things like mana cooldowns and defensive cooldowns can affect the ease and difficulty of the bosses you're taking on.
One reason I can think of as to why your raid leader is extremely reluctant to mix classes is to maximize gear drops. Two paladins in a raid? That means twice as much plate gear needs to drop to outfit you both. The thing is that this would only be applicable after a gear reset (like a new expansion or a new tier). Really though, at this stage of the game that isn't even necessary. Going to PUGs to solve healing problems with the raid seems to be an extraordinary solution.
As a healing leader, the only time I would make the call to my raid leader to not bring a healer from the guild is if we specifically needed some type of buff that we didn't have in the raid. This buff would have to be a make-or-break type of deal. It would be on the level of a difference between a 2% wipe or a kill. Now if your raid leader isn't bringing guild healers because said leader doesn't have confidence in them, that's one thing. Maybe the healers aren't good enough for the stuff your raid is working on and he's rolling the dice on outside healers. Who knows?
How to change their mind
This is tough. If leaders have been doing something traditionally for a long time (such as not diversifying healing classes and bringing them to raids), then it'll be difficult to change their mind. It would be easier if your raid leader was a healer, as he would have a better understanding of the capabilities of each class.
I'd say to bring the topic up to them in private. Find out exactly what is preventing them from relying on guild healers. Is it a class problem? Is it an experience problem? Are some of the healers experiencing high lag that's affecting their ability to heal and do stuff? Is it the belief that certain healers can't work out the raid mechanics? This is a problem that has to be isolated because it's really going to cut into your guild from a morale standpoint. It won't be long before talk of "Why is the boss bringing healers outside of the guild into the raid?" starts permeating.
Heck, I'd be surprised if that kind of talk hasn't happened already. Actually, if it has happened, that's good. It gives you a great opening to start the conversation and figure out what is motivating the raid leader to make those calls in the first place. If your raid leader doesn't feel comfortable or confident in the existing guild healers, that's something that needs to be addressed internally. Find out why that's the case and what needs to be done to address it (even if that includes letting players go or sitting them on the sidelines and looking for other healers who can deliver what is needed). Ideally, those healers can be coached and have their mistakes fixed quickly.
If the reason the raid leader is pulling PUGs in because they're not sure about the actual capabilities of the different classes, then it sounds to me like they could use a bit of education or experience. There is absolutely nothing wrong with bringing two of the same healing class to a raid. Bringing three might be a bit of a stretch though (unless they're all priests).
Anyway, that was a solid question to ask. I hope my thoughts have helped.
Need advice on working with the healers in your guild? Raid Rx has you covered. Send your questions about raid healing to firstname.lastname@example.org. For less healer-centric raiding advice, visit Ready Check for advanced tactics and advice for the endgame raider.