I would like to know where in Auckland can I buy a sapodilla plant or the seeds or the fruit?
We don't have access to that kind of information. You will need to contact your local nurseries or do an online search.
Tried squeezing a few flowers but not fruitful.
This is common in Sapodilla trees that are not quite mature enough to bear fruit. They will flower quite often, but will not bear fruit until they are old enough to do so. This can take about 5 - 8 years.
Here is an article that will help you:
https://www.gardeningknowhow.com/edible/fruits/sapodilla-trees/sapodilla-fruit-dropping.htm
My chikoo(sapodilla/sapota) tree is not giving flowers and if it gives one or two they turn black. What may be the issue?
Unfortunately, your photos did not come through. I am unable to see the damage in question. This could happen for a few reasons, though.
Wet soils, or tightly packed soils, will lead to disease. This can lead to improper nutrient uptake, as well. The additions of earthworms and mulch can help with this.
Another thing to consider will be how long it takes for them to fruit properly. It can take up to eight years to see any fruit production from the tree. If it is still young, then I would give it more time.
Here is an article for more information:
https://www.gardeningknowhow.com/edible/fruits/sapodilla-trees/sapodilla-fruit-dropping.htm
Is this a new bud in sapota/sapodilla?and is it alive?
Yes, and it appears alive to me! Living leaves and green will indicate that the plant is alive and well.
This article will help you with its care:
https://www.gardeningknowhow.com/edible/fruits/sapodilla-trees/growing-sapodilla-fruit.htm