Is there an idiom or expression for that "squeezing sensation you feel in your chest" when you contemplate one you love

Solution 1:

It tugs on the heartstrings:

Used in reference to one's deepest feelings of love or compassion.

‘the kitten's pitiful little squeak tugged at her heartstrings’

‘Ned felt something tug at his heartstrings, sympathy overwhelming him.’
_{Oxford Dictionaries}

Like "heartache" and "broken hearts" it doesn't have to refer to a physical sensation, but it can.

Solution 2:

There are a few different idioms that I suspect were inspired by this feeling, but most of them are used so frequently for the emotions that accompany or inspire the physical sensation that we take them more as metaphors than as physical descriptions.

For example, we often say that our heart swells with happiness or love or pride (MacMillan Dictionary). Though this could mean that our metaphorical "heart" is, metaphorically, expanding from an influx of positive emotion it also seems like a pretty good descriptor for an "expansive or compressing feeling we feel in our chest". The similar expressions "heart-full" or "[somebody's] heart is full" could spring from the same feeling.

Tug or pull or pluck the heartstrings also seems likely to have a physical origin, especially since the term "heartstrings" itself comes from an early (mis?)conception about the anatomy of the heart (Etymonline).

Personally, I call this a zing. It isn't quite as common as the two expressions above, but it has been used for the physical description. It sometimes appears in conjunction with "heartstrings", as in two Judy Garland standards, "Zing! Went the strings of my heart" (Youtube) and "The Trolley Song" (also Youtube):

Clang, clang, clang went the trolley
Ding, ding, ding went the bell
Zing, zing, zing went my heart strings
From the moment I saw him I fell

I haven't found this exact definition for zing, but examples of its use abound:

My feelings are felt in my chest – a pang in my chest when I may be doing the wrong thing, a vibrant zing when my body approves, guiding me at times when logic fails.

—Yvette V. Lapayese, Mother-Scholar: (Re)imagining K-12 Education, 2012, quoting a study participant's journal

A delicious zing of electricity rocketed around in my chest, and I smiled. . . . The zing came again, but this time it felt more tender. For a second I was worried I might cry. But then it passed.

—Katherine Howe, Conversion, 2014

After seven months of marriage she still got a zing through her chest every time he smiled.

—Jennifer Lee, A Wild Goose Chase Christmas, 2012