I would suggest you ask the union rep and Human resources rep about this in a meeting and ask what might be the basis for making this claim and what you need to do to make it possible for her to attend the meetings. You might need to meet first with your HR rep before inviting the union rep to the meeting.
Once you have agreement between the Union and HR, I would invite the rep and the employee to a meeting with HR to discuss what can and can't be required of the employee in terms of meetings and professional development. If the Union rep and HR are in a agreement as to the things the person can refuse to do, then this meeting should make it clear to the employee what her limits are and what will get her in trouble as far as performance. Having the union rep at the meeting already would take away her excuses.
If on the other hand, the union rep says this behavior is ok (and please get him or her to cite the relevant contract clauses), get the union rep to tell you what you can do to remove the obstacle. Do you need to have a third person present? Do you need to replace in person meetings with emails? What? Again once you know what you can ask of the employee, have a meeting with all the relevant parties.
When you end up meeting with the employee, see if you can find out why she objects to one on one meetings. Is there something you can do to make her more comfortable?
Follow up your meeting with a written confirmation of what was said and what the employee will be expected to do to be considered in compliance. Reference this document if she continues to refuse and then use it to start the process of documenting her performance issue (refusing to do valid work related tasks is a performance issue).
Just because she is in a union doesn't mean she can't be fired for cause, it means you have to carefully document what you do and don't skip any steps along the way. Your HR should be familiar with the necessary steps that the union has agreed to.