With preg_match_all:

preg_match_all('~^> .*(?:\R> .*)*~m', $txt, $matches);
$result = $matches[0];

(where \R is an alias for several newline sequences)

With preg_split:

$result = preg_split('~^(?!> ).*\R?~m', $txt, -1, PREG_SPLIT_NO_EMPTY);

that splits the string on each line that doesn't start with > . To trim the newline at the end of each block, you can start this pattern with an optional \R? => ~\R?^(?!> ).*\R?~m or like that ~(?:\R?^(?!> ).*)+\R?~m to eventually grab several lines at a time.

About \R:
\R is by default an alias for (?>\r\n|\n|\x0b|\f|\r|\x85) (any non-utf8 8bits characters sequences for a newline). In utf8 mode, with the u modifier or starting the pattern with (*UTF8)(*BSR_UNICODE), two other characters oustide of the ASCII range are added to the list: the line separator (U+2028), the paragraph separator (U+2029).
It's handy when you don't know which newline sequence is used in the string but slower than writing the exact newline sequence if you know it. You can restrict \R to (?>\r\n|\n|\r) with the directive (*BSR_ANYCRLF) at the start of the pattern.

My idea is to split the string based on the line breaks. maybe this will help you?

foreach(explode("\n", $string) as $key=>$val) {
  if(preg_match('/^(>.*)$/', $val, $match))
    echo $match[1] . PHP_EOL;
}

output:

> Don't forget to buy eggiweggs on the way home.
> Also remember to brush your shoes.
> > > And clean up after the pigs.
> > But I have no pigs.
> Yes, you do. Your kids.

edit: i tried something else... but it is not perfect

preg_match_all("/(>[^\n]+)/sm", $string, $match);
print_r($match);

output

Array
(
    [0] => > Don't forget to buy eggiweggs on the way home.
    [1] => > Also remember to brush your shoes.
    [2] => > > > And clean up after the pigs.
    [3] => > > But I have no pigs.
    [4] => > Yes, you do. Your kids.
)

Explicitly match the end of the quote

Rather than attempting to tweak greedy/none-greedy behavior it's easiest to solve regex problems if you can match something unambiguous at the start/end of the match.

When a quoted block continues on multiple lines, the last characters of the line are [newline] + >, when a quoted block ends the last characters of the quote are [newline] + [not >] - This logic/pattern allows finding the whole quoted blocks.

From this logic we arrive at:

/(> .+?)\n[^>]/s

This is:

/           # Regex start delimiter
  (         # Start of capturing group
    >       # Literal >
            # Literal space
    .       # Any character, including newlines
    +?      # At least once, none-greedy match
  )         # End of Capturing group
  \n        # Newline
  [^>]      # anything _except_ a literal >
/           # Regex end delimiter
s           # PCRE_DOTALL flag (makes . also match newlines)

The use of a none-greedy match here prevents the regex matching the start of the first quote all the way to the end of the last quote.

Choosing between (or combining) PCRE_DOTALL and PCRE_MULTILINE depends on the strategy employed - here the intent is only to modify the behavior of .. More info in the docs.

If the source text is coming from windows, you may wish to use \R (as noted in a different answer).

Demonstration

<?php

$input = <<<STUFF
> Don't forget to buy eggiweggs on the way home.

I shall not.

> Also remember to brush your shoes.

Will do.

> > > And clean up after the pigs.
> > But I have no pigs.
> Yes, you do. Your kids.

I see what you mean. They sure make a mess.
STUFF;

$regex = "/(> .+?)\n[^>]/s";
preg_match_all($regex, $input, $matches);

print_r($matches);

Results in:

Array
(
    [0] => Array
        (
            [0] => > Don't forget to buy eggiweggs on the way home.


            [1] => > Also remember to brush your shoes.

            [2] => > > > And clean up after the pigs.
> > But I have no pigs.
> Yes, you do. Your kids.


        )

    [1] => Array
        (
            [0] => > Don't forget to buy eggiweggs on the way home.
            [1] => > Also remember to brush your shoes.
            [2] => > > > And clean up after the pigs.
> > But I have no pigs.
> Yes, you do. Your kids.
        )

)

Why don't the attempts in the question work?

So if I do: #^(> .+?)$#us

The s modifier only affects ^ and $.

This regex is anchored to the start and end of each line (does a none-greedy match, but the . will not match a newline anyway) - hence it matches each quoted line individually.

And if I do: #^(> .+?)$#um

The m modifier only affects ..

It has no effect on ^ or $ - so as noted in the question this can at most produce one match.

Flags are not mutually exclusive, and can be used in combination.