An integer value of 0 - zero - is not rendered by default with the widget badge because it is rated as 'nothing'.

Since due is computed and not stored anyways, you may as well just compute a second field and convert it to a string:

# -*- coding: utf-8 -*-

from odoo import models, fields


class ProductTemplate(models.Model):
    _inherit = 'product.template'

    expiry_date = fields.Date(string='Expiry Date')
    due = fields.Integer(compute='_compute_due')
    due_label = fields.Char(string='Days Left', compute='_compute_due')

    def _compute_due(self):
        for record in self:
            record.due = (record.expiry_date - fields.Date.today()).days if record.expiry_date else 0
            record.due_label = str(record.due)
            # record.due_label = str(record.due)

... and use it as:

<?xml version="1.0" encoding="UTF-8" ?>
<odoo>
    <record id="product_template_tree_view_inherit" model="ir.ui.view">
        <field name="name">product.template.product.list.inherit</field>
        <field name="model">product.template</field>
        <field name="inherit_id" ref="product.product_template_tree_view"/>
        <field name="arch" type="xml">
            <xpath expr="//field[@name='name']" position="after">
                <field name="expiry_date"/>
                <field name="due_label"
                       decoration-danger = "due &lt;= 0"
                       decoration-warning = "due &lt;= 30"
                       decoration-info = "due &gt; 30"
                       invisible="not expiry_date"
                       widget="badge"/>
            </xpath>
        </field>
    </record>
</odoo>

... which will result in:

You can simply recycle _compute_due() fort this, as its doing the same thing with the same values anyways too. 

Also, with an actual string label you could do so many more shenanigans, such as:

    def _compute_due(self):
        for record in self:
            record.due = (record.expiry_date - fields.Date.today()).days if record.expiry_date else 0
            if record.due == 0:
                record.due_label = _('expiring today')
            elif record.due > 0:
                record.due_label = _('%s day' % record.due) if record.due == 1 else _('%s days' % record.due)
            else:
                over_due = abs(record.due)
                record.due_label = _('%s day overdue' % over_due) if over_due == 1 else _('%s day overdue' % over_due)

On another note, however: you are aware about the fact that Odoo actually has  an expiry management integrated, right? https://www.odoo.com/documentation/applications/inventory_and_mrp/inventory/product_management/product_tracking/expiration_dates.html