As someone that works in support for a hosting provider, you're the type of customer that irritates me the most. While they shouldn't be rebooting your box to get root access without your consent, you should at least help them help you. Give them an account with limited sudo access to view your logs. If that won't do, then provide them with the necessary logs. If that's not good enough, don't expect support and move your stuff to some place that doesn't provide the level of support you're paying for.