Not at all surprising to me that the 4x4 "mast" broke as you describe. The geometry of a zip line is such that the tension on the line is many times greater than the weight of the rider.
One secondary source suggests installing the cable such that the sag is 5% of the span. If this guideline is followed, the weight of the rider creates a pulling force at each cable support of 5 times the riders weight. The support also needs to take the load created by the cable itself, which is also a multiple of the weight of the cable.
Apparently the Association for Challenge Course Technology (ACCT) and the Professional Ropes Course Association (PRCA) both publish standards on zip line design and installation, which should be considered the authority on such matters.
DISCLAIMER Everything that follows is illustrative only, the above references should be consulted before building a zip line.
So now to attempt to answer your question: How many pounds of "pull" should a piece of 4x4 lumber hold, given a load applied 3 feet from the support point?
You've essentially got what's called a cantilever beam. In a cantilever beam you have both bending and shear stresses to consider, though in this case the bending stresses are the primary concern.
A #2 southern pine 4x4 post has a maximum allowable fiber stress of 850psi.
For a simple beam of square cross-section
Max bending stress = 6*M/(b^3)
where M is the bending moment (load in the cable (P) times the overhang of 36 inches)
and b is the length of a side of the square (3.5 inches for a 4x4)
Rearranging a bit to solve for the max "pull":
P=(b^3*max allowable stress)/(6*length of overhang)
P=(3.5in)^3*850psi/(6*36in)=168lb
So the post can withstand about 168 lb of pull from the cable (really it should be a little bit less when the shear forces are also considered).
But as I stated earlier, the tension in the cable can be around 5 times the weight of the person hanging from the zip line, so the max "safe" weight with this back-of-envelope calculation is only around 168lb/5=34lb, even before applying a typical factor-of-safety to the calculations to account for the dynamic nature of the load, and the fact that a person's (relative) safety depends on it.
So based on these calculations, even with just kids on the zip line, you were very near, if not exceeding the design load limits for your 4x4 post.
With a 3 foot overhang, a 4x4 post is nowhere near strong enough to safely support a zip line, even for kids.
According to the table in the 2012 IRC code a exterior deck needs to handle a 60lb per sq ft (40 lb live + 20lb dead load). Most treated lumber that I am aware of is #2 southern yellow pine, using that criteria on the table in the code book, your joists can carry a 10'-9" span.
The biggest failure in deck construction has not been the posts or the joists, but the connection to the house. I believe code now requires through bolts into the band joist, or the joist of the house, or posts at the house is cool too.
The length of the post will dictate whether they need to be 4X4 or 4X6. Its a deflection (bowing) thing. IMO 4X4 for a 10' span is not overloading the post, a 16' span may...
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To answer correctly, you would need to show or say the dimensions of the tub and the distance from the existing 4x4 supports. Don't forget the additional weight of 2 people. If it is near a support post, you will be fine. I would play it safe and add a post, secured to a 2x6 directly under the tub.