Whenever I see cute babies or animals, I oftentimes feel like pinching them, not to hurt them but to express how much I find them cute, fluffy, chubby, or adorable.
What word or short phrase can I use to describe that feeling?
There is the (somewhat specialist) term cute aggression.
From Scientific American, July 2013:
New research by two Yale University psychologists details how the sight of something cute brings out our aggressive side. Rebecca Dyer and Oriana Aragon investigated “cute aggression” by showing study participants slide shows of either cute, funny or normal animal photographs. As they watched, the participants held bubble wrap. The researchers, attempting to mimic the common desire to squeeze cute things, told subjects to pop as many or as few bubbles as they wished. People watching the cute slide show popped significantly more bubbles than those viewing the funny or control pictures, according to results presented at the Society for Personality and Social Psychology annual meeting in New Orleans. “Some things are so cute that we just can't stand it,” Dyer concludes.
Cute aggression's prevalence does not mean that people actually want to harm cuddly critters, Aragon explains. Rather the response could be protective, or it could be the brain's way of tamping down or venting extreme feelings of giddiness and happiness.
As you might expect, the term and the research have garnered considerable media interest, so much so that there is no shortage of journalism out there on the topic - and even a Wikipedia article which notes that later researchers have used the term playful aggression.
However, appropriate as they may be, neither "cute aggression" nor "playful aggression" have particularly wide currency, nor, being technical and specific, are they particularly cute terms, as one might feel they should be; you can't just throw them around in conversation and they lack any warmth or emotion.
Dreadful a word as it now is - and though it lacks the requisite specificity - squee might better serve to describe the cuteness overload you are experiencing. The web-available Oxford describes "squee" as:
Exclamation, informal: Used to express great delight or excitement
giving the example sentence:
Those chairs are there for moms and bridesmaids to sit on and SQUEE when the bride-to-be emerges from the dressing room.
(It is worth noting that the capitalisation in that example is in the original!)
"Squee" has connotations of high-pitched loudness, excitedness - and there is some phonic association with "squeeze" - although the above dictionary reference says it isn't related etymologically.
Given that this relatively obscure term has exploded of late, Urban Dictionary is worth looking at:
Verb: To squeal with glee; from a combination of the two words; the sound of an excited fangirl.
The folk etymology, although inaccurate, does convey its meaning well. (See also the "squee" answer in this question, Is there a name for the emotional response to cuteness?)
"Squee" is far from perfect. Aside from the technical terms, it's the best I got, though.
(Really interesting ngrams, for what it's worth - why the WWII-era explosion of capitalised 'Squee'? ...though I don't think its current vogue has translated into any real impact on the presses yet.)
P.S. After writing this, before submitting my answer I followed one of the links in the Wikipedia article to a piece in Lateral Magazine, which reassured me in my suggesting "squee":
Although the term "cute aggression" did not exist in the English language until recently, many other languages do have words to describe these strangely aggressive feelings towards something cute. In Bahasa Indonesian, "gemas" translates into English as "something so cute you want to do violence to it". The Czech word "muchlovat" means the desire to squeeze a cute person. In the Tagalog language, spoken in the Philippines, “gigil” describes the gritting of your teeth and the urge to pinch or squeeze something that is unbearably cute. These translations all capture that "sqeeeeee" feeling better than any word I know.
You're a pincher.
Dictionary.com - pincher
a person or thing that pinches.
Merriam Webster - pincher
one that pinches
Chamorro is the native and spoken language of the Chamorro people who are the indigenous people of Guam and the Northern Mariana Islands, US territories.
Chamorro-English Dictionary - Ma’goddai: excited, angered, fighting mad.
Guampedia.com - Ma’goddai
"The Chamorro term ma’goddai can be explained as a strong feeling one gets when they admire someone because of their appearance such as being poki (pleasantly chubby), cute, and having unique traits that distinguishes him or her from others. This causes an urge to pinch, squeeze, bite or smother that person with kisses, particularly a child. It is believed to date back to ancient society and is still practiced today. Ma’goddai has significance for a person’s well being. It is believed that if a person refrains from following through on his or her urges, then the person that making them feel ma’goddai will become ill.".