[H]ow do I convey and drive home the point that an employee's personal health and emergencies are more critical and not to sacrifice those for the sake of achieving deadlines?
First, lead by example. Honest and conscientious managers are the hardest and longest working members of the team and telling your team to take it easy when you are regularly getting in early, staying back and working on weekends may seem hypocritical.
For example, if you have a personal emergency, prioritize it over work. More regularly, instead of constantly staying back late, leave at a reasonable time and finish up the work from home. Alternatively, be seen to be leaving, have a coffee then come back to work (usual for Japan where team members will not leave before their boss).
Second, when people say "work like balance", they usually mean flexibility. Often people are happy to do the amount of work required. They just cannot do it from 9 to 5 at the office all the time, particularly if they have a family or other responsibilities.
If you can, give people the ability to work from home via VPN. Allow people to start early or work late. If people want to get in early then go to the gym during the day, that's fine. If this is a regular occurrence, it is less of an event when something more serious happens. Think of it like a fire drill for personal emergencies.
Third, communicate. As you are doing, make sure team members can talk to you if they need help. Share outcomes of the discussions (not personal details) with the team. For example, make sure to tell the team that the team member has a personal issue and will be back working the following week. This creates a zone of safety.
Involve senior members of the team in this, too. You may not he available all the time so make sure the people that team members go to when you are not around know.
Fourth, your remuneration and promotion incentives or culture may encourage quality or perception of work over quality. Bell curve systems, where only the top performers are rewarded, are particularly bad for this. As a manager, you may have limited control but make sure to defend to upper management if questions arise.
Fifth, manage competitiveness within or between teams and any commitments outside the teams' core responsibilities. These can create lots of hidden additional work over the teams' regular work.
All that said, remember that it is up to individuals to manage their own lives. Some may really enjoy working (developers are particularly bad at this) or want a promotion.
As a manager, the key is to look for the signs when people are overextended, such as prolonged tiredness, irritability and frustration. In this case, you need to raise it a way that does not sound like a punishment or admonishment ("Why did you yell at Sarah? Have you been working too late again? You know you work too hard. Can't you limit yourself like everyone else?") and more a shared empathy and goal ("It has been really tough meeting those deadlines, hasn't it? I just want you to know that everyone realizes how much effort you are putting in. Just remember that to let me know if you need a hand. If something happens to you, we are all screwed.") If you handle these types of issues well, chances are personal problems will be a lot easier.