My supervisor has suggsted someone as an external examiner and I am a bit worried that they are not suitable. They have published a lot on the subject matter that my thesis deals with but they use a different (and competing) approach to analysis. Does this matter? I would hate for them to say that my approach was wrong and that I had to redo everything!

I think the usual way is to have one examiner that's familiar with your topic, and one that's familiar with your method. So it might be that your internal is more familiar with your approach.

Discuss it with your supervisor and see if they can prep you for any challenging questions that the examiner might have.

Thanks!

An additional question to this topic - next week there is a seminar on at a nearby university and there is someone presenting who, on paper, sounds like a good fit as an external examiner. I am going to the seminar anyway but just wondered if I should introduce myself with an eye to inviting her to be my external, if appropriate? Or is this not the done thing?

Definitely introduce yourself, but don't ask her to be your external. There is usually a formal procedure involved where the university writes to your external to ask them so it would be unprofessional for you to ask her.

If you get a feel for whether or not she's interested in your research, you can mention that to your supervisor.

I don't know if you realise but external examiners get paid very little (usually c. £150 and then taxed as a second income at 50% i.e. a lot less than minimum wage given the time commitment) for examining. Your supervisor will be thinking therefore who has the expertise and a) owes me a favour and might therefore shoulder the extra work to examine my student and b) who is known as a decent and fair person. The person who looks great on paper might be known as not being very nice in real life. Your university may also have specific requirements for an examiner i.e. number of years of experience, employed at research intensive university, based in the UK to cut expenses etc.
And Butterfly's right - don't ask people (especially strangers) to be your external - it will send a signal that you don't know how the system works or possibly even worse that you're trying to fix the examination process. That's not the professional reputation you want!

Thanks don't worry, I wasn't going to ask her outright! It's great to hear a bit more about the system though, as I am never on campus, I really don't know how these things work.