To Push or not to Push?
To push or not push? Consider your child's felt sense of safety and explore root causes.

There are times when we are not sure how far to push our children when they're facing a challenge. Are they showing big emotions about the challenge because they really can't handle it, or just because they're having emotions about it (and that's OK)? Do we use 'tough love' or take a more gentle approach?
The kinds of challenges I am thinking of are those where the child is mandated to participate, and/or where it has been for their well-being. Some examples are; anxiety about going to school resulting in total refusal to go to school, trouble with a school subject (or teacher) resulting in absolute refusal to even try doing the work, dislike for a sport or sport's coach and refusing to participate, and giving up on a hobby they have always been passionate about.
Rule out harm
When faced with these types of situations where children refuse to continue participate, we must always first rule out any harm or abuse. If the refusal is around someone or something that makes them feel unsafe and/or has actually harmed them, then that is obviously of utmost priority to be dealt with.
If it is clear there is no harm being done to the child, then we are again faced with the question: How far do I push?
I don't have an (or "the") answer for you. Knowing when to push your child and when not to is dependent on so many factors including the uniqueness of the child themselves, and the situation.
Try a little push
I will say that perhaps it is OK to push a little in most cases and assess the child's reaction. In many cases when we show our children we believe they can do it, and provide a firm and kind approach, they settle. For example, children going to pre-school for the first time need some settling in time and firm and kind is most often the way forward.
Sometimes we get overwhelmed by our children's big emotional reactions to things, and we can feel like we don't know what to do. This can cause us to shy away from pushing at all.
Remember that emotions are communication. Big emotions can certainly be confronting, loud and even scary, yes. But they are not going to kill the child or us. It may sometimes feel "end of the world-ish" and it is OK if you feel at the end of yourself or unable to cope. It can be a lot.
If pushing creates a meltdown (a meltdown is a an experience of intense emotional overwhelm and is not within the person's ability to control), then you know you cannot continue to push in that moment. They are simply unable to cope with the emotional reaction of what they're being asked to do.
It can be very helpful to have professional help and say "yes" to help offered from family/friends/teacher that you trust and whose values resonate with yours. At the same time, be aware that if you get advice from too many people, it will start to feel very murky and confusing.
Am I safe?
At the root of the child's refusal to engage is a perception on their part that they are not safe; in that moment, or that scenario, or relationship. They cannot trust themselves to be or feel safe and so they resist. We are all hard-wired to protect ourselves from pain. Feeling unsafe is painful, and causes deep discomfort.
When we can understand what the perception is that they are carrying, whether it is real or imagined, we can help the child start to deal with that perception and thus hopefully restore a neuroception of safety again.
This is not always a simple matter and asking a child "Why are you feeling this way?" does not always help. They often simply have no way to put into words what their fears are about. They may give you answers, but these may simply be excuses rather than the true root cause. In my experience, when a child gives you a different answer each time you ask "why", that they are not able to articulate the true reason that they are feeling unsafe.
Help create a neuroception of safety
As a parent, you can help a child feel safe by preparing them for what is to come. Many parents have an excellent idea of how far in advance to prepare their particular child. Some know they need a week's notice and others need an hour. Make the "unknown" known where it pertains to new experiences and people.
Creating a sense of safety may actually happen with someone other than a parent. Sometimes a trusted adult just one step removed from the child can be effective at helping the child to feel safe. Another adult who doesn't feel all the complicated feelings and anxieties the parent may be feeling may be able to help the child navigate the experience and help them settle.
Praise and affirm even tiny steps forward that they child takes. Talk about what it felt like for them and praise their effort.
Dig deeper
You can try everything and still not see a shift. You may need help digging deeper. Are there any traumas or losses that have occurred that have interrupted the child's felt sense of safety? Even overhearing adults talk about a scary event can trigger fear and anxiety. Consider also the emotional temperature of your home environment. If it is fraught with conflict and stress, this will have a knock-on effect on children. If this is the case, definitely seek professional help. The anxiety and feeling of being unsafe won't stop until there is more emotional stability in the home.
Sometimes, this is not an easy or simple journey. Patience and consistency are key.