What is the best way to achieve a more natural looking breast that jiggle when moving/jumping and feels more squishy? (photo)

OK, no sugar coating.  Here are my suggestions.  You definitely need a vertical mastopexy to lift your areolas and reduce their size.  This will also reduce your space between your breasts, but the space will always be wider than ideal but acceptable.  The implants need to go under the muscle.  They need to be silicone and smooth and moderate plus round profile would probably be perfect.  The softness will be great unless you suffer from a capsular contracture after the surgery.    You have a tough case but with excellent surgical technique, it should be a fabulous result.

Thanks for your question.  If you are looking for smaller areolae and a higher position of the areolae on the breast mound, you might want to consider having combined breast lift plus implants surgery.  Your surgeon will be able to discuss all of the options for this procedure.

Choosing breast implants and their placement is a detailed process that requires an in-person examination and trying out sizers to see which you like better.  it cannot be done on the internet if you expect actually good results.  Be sure to discuss with your board certified plastic surgeon exactly what your goals are, and how reliably they can be expected to be achieved.

Hello and thank you for your question.  The
best advice you can receive is from an in-person consultation.  With that
being said, based on your photographs, the best shape and contour could be
achieved with an implant and a wise pattern breast lift. Your nipple sits
below the inframammary fold and without a lift in addition to the implant, this
will not have a natural shape.  A wise pattern lift with an implant can
result in a very nice shape if good technique is used.  The size,
profile, and shape of the implant is based on your desired breast
size/shape, your chest wall measurements, and soft tissue quality. 
This decision should be based on a detailed discussion with
equal input from both you and your surgeon.   Make sure you
specifically look at before and after pictures of real patients who have
had this surgery performed by your surgeon and evaluate their
results.   The most important aspect is to find a surgeon you are
comfortable with. I recommend that you seek consultation with a qualified
board-certified plastic surgeon who can evaluate you in person.
Best wishes and good luck.
Richard G. Reish, M.D.
Harvard-trained plastic surgeon

Dear lunaluv601,First and foremost you need a breast lift.  You would then require either breast implants or fat transfer to replenish the lost volume in the upper breasts and create your cleavage.  The areolae would be reduced with the breast lift.  I think jiggly and squishy should be your least concern.  Good luck!

I prefer to place them under the muscle. Silicone are more preferred by my patients.  Your cleavage will be limited by your lack of cleavage and your anatomy.  

Hello and thank you for the question and the photos. Your photo shows sagging or ptosis of the breast. I would suggest a full mastopexy using the anchor type incision is the best option. This will give you the best ability to lift and reshape the breasts. During the surgery, the nipple/areolar diameter will be made smaller as well. 
The surgery takes about 2-3 hours under full anesthesia. It is performed in a fully licensed and accredited outpatient surgical center. 
I would normally recommend patients taking 1 - 2 weeks off of work and about a month out of the gym.
The gap between the breasts is based on the size shape of your sternum ( breast bone ). This cannot be changed. The implants however will help minimize the appearance of the gap. Silicone implants do feel the most natural and is typically done using the dual plane technique. A soft feeling breast is always the goal. saline implants tend to feel firm compared to gel. Taking steps to avoid a capsular contracture ( firm scar tissue) around the implant is key to long term positive outcomes. 
You should have several consultations with board certified plastic surgeons in your area before proceeding. 
As always it is best to be healthy, no smoking, and if you have any health problems they are under the current management of your primary care physician.
Best to you