Should you Walk like a Penguin 

for safety in the winter?

This meme is widespread, with mostly good suggestions:
penguin walk tips
  Footwear is also important: notice that penguins have natural spiked crampons on their feet!

You cannot walk in a normal manner and be safe on slippery ice. 
The "walk like a penguin," advice is useful in that it makes you think more about how you are walking, resulting in slower, more careful ambulation.  But shuffling along with your head bent down and weight forward is not always a good solution.

This very popular infographic is somewhat misleading in its analysis of the forces involved:
penguin walk infographic

The problem is not that your weight is split in mid-stride, it is that your forward momentum puts much more force in a forward horizontal direction when the front foot strikes the ground.  Normal walking is quite efficient: on a flat path we recover about 65% of our energy by planting the front foot and letting our momentum carry us forward and over it.  In a normal walking gait, every step is actually a forward fall, caught by our front foot as shown below.
normal walk

But, if there is no friction to hold the front foot it shoots forward, and driven by momentum and your weight pressing into the skidding leg, down you can go.  In the image below, a front mis-step and slip is simulated:

walk and slip

Ideally, you can use "Tai chi walking" to reduce the horizontal slipping force to nearly zero.  You balance on one leg while the other reaches out "empty" with no weight.  When the front foot touches the ground you smoothly transfer your weight from the back leg to the front leg: the force in the front foot is almost straight down.  When all your weight is on the front foot the empty back foot is picked up and gently moved to the front, placed on the ground, and the cycle repeats.  This image below shows the slow and exaggerated stepping movement from the Tai chi form.  Although the step is large, there is no forward push on the front foot, or rearward push-off from the back foot.

taiji walk

This may sound complicated but you did it all the time as a kid: "walking tiptoe" we called it.  Whether you touch with the toe or the heel first the important thing is that there is hardly any weight on the foot when it makes contact.

To walk more safely you must give up some efficiency and speed:  reduce your momentum by walking more slowly and shorten your stride length to shrink the horizontal force pushing forward on the front foot.  Step forward lightly and carefully.

Be sensible, pay attention, and shift into more of a "Tai chi walking" movement when crossing slippery or unsure  areas.

But, don't drag your feet by shuffling, or bend your head over to look straight down. 

Shuffling along with the feet close together and the center of gravity (your weight) always forward is a recipe for a forward tripping fall.  This is true even for penguins!  As a study by Dr. Astrid Willener while at Roehampton University in London found:

penguins at shore

penguin cgm

fat penguin

   A fully-stuffed penguin walking in
   from just feeding.

Lastly, remember that when penguins face a flat
and really slippery surface, they have other options!

penguin bellyflop
  From Ranger Rick's nature magazine